E
NERGY AND
E
NVIRONMENT
Volume 4, Issue 1, 2013 pp.93-102
Journal homepage: www.IJEE.IEEFoundation.org
Exergoeconomic performance optimization for a
steady-flow endoreversible refrigeration model including six typical
cycles
Lingen Chen, Xuxian Kan, Fengrui Sun, Feng Wu
College of Naval Architecture and Power, Naval University of Engineering, Wuhan 430033, P. R. China.
Abstract
The operation of a universal steady flow endoreversible refrigeration cycle model consisting of a constant thermal-capacity heating branch, two constant thermal-capacity cooling branches and two adiabatic branches is viewed as a production process with exergy as its output. The finite time exergoeconomic performance optimization of the refrigeration cycle is investigated by taking profit rate optimization criterion as the objective. The relations between the profit rate and the temperature ratio of working fluid, between the COP (coefficient of performance) and the temperature ratio of working fluid, as well as the optimal relation between profit rate and the COP of thecycle are derived. The focus of this paper is to search the compromised optimization between economics (profit rate) and the utilization factor (COP) for endoreversible refrigeration cycles, by searching the optimum COP at maximum profit, which is termed as the finite-time exergoeconomic performance bound. Moreover, performance analysis and optimization of the model are carried out in order to investigate the effect of cycle process on the performance of the cycles using numerical example. The results obtained herein include the performance characteristics of endoreversible Carnot, Diesel, Otto, Atkinson, Dual and Brayton refrigeration cycles.
Copyright © 2013 International Energy and Environment Foundation - All rights reserved.
Keywords: Finite-time thermodynamics; Endoreversible refrigeration cycle; Exergoeconomic performance
1. Introduction
Recently, the analysis and optimization of thermodynamic cycles for different optimization objectives has made tremendous progress by using finite-time thermodynamic theory [1-14]. Finite-time thermodynamics is a powerful tool for the performance analysis and optimization of various cycles. For refrigeration cycles, the performance analysis and optimization have been carried out by taking cooling load, coefficient of performance (COP), specific cooling load, cooling load density, exergy destruction, exergy output, exergy efficiency, and ecological criteria as the optimization objectives in much work, and many meaningful results have been obtained [15-27].
bound at maximum profit was termed as finite time exergoeconomic performance bound to distinguish it from the finite time thermodynamic performance bound at maximum thermodynamic output.
There have been some papers concerning finite time exergoeconomic optimization for refrigeration cycles [31, 33, 38]. A further step in this paper is to build a universal endoreversible steady flow refrigeration cycle model consisting of a constant capacity heating branch, two constant thermal-capacity cooling branches and two adiabatic branches with the consideration of heat resistance loss. The finite time exergoeconomic performance of the universal endoreversible refrigeration cycles is studied. The relations between the profit rate and the temperature ratio of working fluid, between the COP and the temperature ratio of working fluid, as well as the optimal relation between profit rate and the COP of the cycle are derived.The focus of this paper is to search the compromise optimization between economics (profit rate) and the energy utilization factor (COP) for the endoreversible refrigeration cycles. Moreover, performance analysis and optimization of the model are carried out in order to investigate the effect of cycle process on the performance of the cycles using numerical examples. The results obtained herein include the performance characteristics of endoreversible Carnot, Diesel, Otto, Atkinson, Dual and Brayton refrigeration cycles.
2. Cycle model
An endoreversible steady flow referigeration cycle operating between an infinite heat sink at temperature
H
T and an infinite heat source at temperature TL is shown in Figure 1. In this T-s diagram, the processes
between 2 and 3 , as well as between 5 and 1 are two adiabatic branches; the process between 1 and 2 is a heating branch with constant thermal capacity (mass flow rate and specific heat product) Cin; the
processes between 3 and 4, and 4 and 5 are two cooling branches with constant thermal capacity Cout1 and
2
out
C . In addition, the heat conductances (heat transfer coefficient-area product) of the hot- and cold-side
heat exchangers are UH1, UH2, and UL, respectively. The heat exchanger inventory is taken as a constant,
that is UH1+UH2+UL=UT. This cycle model is more generalized. If Cin, Cout1 and Cout2 have different values,
the model can become various special endoreversible refrigeration cycle models.
Figure 1. T-s diagram for universal endoreversible cycle model
3. Performance analysis
According to the properties of working fluid and the theory of heat exchangers, the rate of heat transfer
1
H
Q and QH2released to the heat sink and the rate of heat transfer QL (i. e. the cooling load R) supplied by
heat source are given, respectively, by
1 2
H H H
Q =Q +Q (1)
. .
1 1(3 4) 1 1( 3 )
H out out H H
Q =m C T −T =m C E T−T (2)
. .
2 2(4 5) 2 2(4 )
H out out H H
Q =m C T −T =m C E T −T (3)
. .
2 1 1
( ) ( )
L in in L L
where m is mass flow rate of the working fluid, EH1, EH2 and EL are the effectivenesses of the hot- and
cold-side heat exchangers, and are defined as
1 1 exp( 1)
H H
E = − −N ,EH2= −1 exp(−NH2),EL= −1 exp(−NL) (5)
where NH1, NH2 and NL are the numbers of heat transfer units of the hot- and cold-side heat exchangers,
and are defined as
.
1 1/( 1)
H H out
N =U m C ,NH2=UH2/(m C. out2),
.
/( )
L L in
N =U m C (6)
where UH1, UH2 and UL are the heat conductance, that is, the product of heat transfer coefficient α and
heat transfer surface area F
1 1 1
H H H
U =α F ,UH2=αH2FH2,UL=αLFL (7)
The COP ε of the cycle is
( )1 ( ) 1
1 2
1 1
H L H H L
Q Q Q Q Q
ε − −
= − =⎡⎣ + − ⎤⎦ (8)
Combining equations (1) - (3) and (8) gives
4 H1 H (1 H1) L
T =E T + −E xT (9)
( )
5 (1 H1)(1 H2) L H1 H2 H1 H2 H
T = −E −E xT + E +E −E E T
(10)
(
1)
1 L ( L H) 1
T=T −a xT −T +ε−
(11)
( )
(
1)
2 L L (1 L)1 L 1 L ( L H) 1
T =E T + −E T =T −a −E xT −T +ε− (12)
where a=⎣⎡Cout1EH1+Cout2EH2(1−EH1) (⎦⎤ C Ein L),x=T T3 L
Consider the endoreversible cycle 1 2 3 4 5 1− − − − − . Applying the second law of thermodynamics gives
( 2 1) 1 ( 3 4) 2 ( 4 5)
ln ln ln 0
in out out
S C T T C T T C T T
∆ = − − = (13)
From the equation (13), one has
1 2 2 1
2 4 5 1 3 0
out out out out
in in in
C C C C
C C C
T T T T T
−
− = (14)
Combining equations (9) - (14) gives
( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 out in
out in out in
C C
L L L
C C C C
L H L L L L L
T a T xT
a xT T E a xT T a T xT
ε= −
⎡ ⎤
− ⎣ − − ⎦− + (15)
( ) ( ) ( ) 1 1 1 1 1 out in out in C C
L L L
L in L C C
L L
T a T xT R Q mC E
E a xT
−
= =
− −
(16)
where ( ) ( 1 2) ( ) 2
1 1 1 1 (1 1)(1 2) 1 2 1 2
out out in out in
C C C C C
H L H H H H L H H H H H
a =⎣⎡ −E xT +E T ⎦⎤ − ⎡⎣ −E −E xT + E +E −E E T ⎤⎦
The required power input P of the cycle is
( )
(
( ) 1)
( ) ( ) 11 1 1
out in out in
C C C C
H L in L L H L L L L L
Assuming the environment temperature is T0, the rate of exergy output of the refrigeration cycle is:
0 0 1 2
( 1) ( 1)
L L H H L H
A=Q T T − −Q T T − =Qη −Qη (18)
where ηi is the Carnot coefficient of the reservoir i (i=1, 2).
So the rate of exergy output of the refrigeration cycle is
( ) ( ) ( ) ( )
{
1 1}
1 1 1 1 2
out in out in
C C C C
in L L L L L L L H
A=mC E η ⎡⎣T a −T xT ⎦ ⎣⎤ ⎡ −E a− xT ⎤⎦−ηa xT −T (19)
Assuming that the prices of exergy output and the work input be ψ1 and ψ2, the profit rate of the
refrigeration cycle is:
1A 2P
π ψ= −ψ (20)
Substituting equations (17) and (19) into equation (20) yields
( ) ( ) ( ) ( ) ( )
{
1 1}
1 1 2 1 1 1 ( 1 2 2)
out in out in
C C C C
in L L L L L L L H
mC E T a T xT E a xT a xT T
π= ψ η ψ+ ⎣⎡ − ⎤ ⎡⎦ ⎣ − − ⎤⎦− ψ η ψ+ − (21)
4. Discussions
Equations (15) and (21) are universal relations governing the profit rate function and the COP of the steady flow refrigeration cycle with considerations of heat transfer loss. They include the finite time exergoeconomic performance characteristic of many kinds of refrigeration cycles.
When Cin=Cout1=Cout2=C (CV or CP), equations (15) and (21) become:
( )
( ) ( 2 2 ) 2(1 ) ( )
L L H L
L H H L H L L H L H L L H L
T E T xT
xT T E E E E xT E E T T E T xT
ε= −
− ⎡⎣ − − + − ⎤⎦− − (22)
( ) ( )
(
1)
( )2 1 1 2 1 1 2 2
H L H
mCE xT T
π= − ⎡ψ η ψ+ +ε− −ψ η ψ+ ⎤
⎣ ⎦
(23)
Equations (22) and (23) are the finite time exergoeconomic performance characteristic of a steady flow endoreversible Otto (C=CV) or Brayton (C=CP) refrigeration cycle.
When Cout1=Cout2=Cv and Cin=Cp, EH1=0, and equations (15) and (21) become:
( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 k
L L L
k k
L H L L L L L
T a T xT
a xT T E a xT T a T xT
ε= ′ −
⎡ ′ ⎤ ′
′ − ⎢⎣ − − ⎥⎦− + (24)
( ) ( ) ( ) ( ) ( )
{
1 1}
1 1 2 1 1 1 ( 1 2 2)
k k
p L L L L L L L H
mC E T a T xT E a xT a xT T
π= ψ η ψ+ ⎢⎡⎣ ′− ⎤ ⎡⎦ ⎣⎥ ⎢ − ′− ⎥⎤⎦− ′ψ η ψ+ − (25)
wherea′ =EH2 (kEL), [ ]
1
1 (1 2) 2
k
L H L H H
a′ =xT −E xT +E T .
Equations (24) and (25) are the finite time exergoeconomic performance characteristic of a steady flow endoreversible Atkinson refrigeration cycle.
When Cout1=Cout2=Cp and Cin=Cv, EH1=0, and equations (15) and (21) become:
( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 k
L L L
k k
L H L L L L L
T a T xT
a xT T E a xT T a T xT
ε= ′′ −
⎡ ′′ ⎤ ′′
′′ − ⎢⎣ − − ⎥⎦− + (26)
( ) ( ) ( ) ( ) ( )
{
1 1 2 1 1 1 ( 1 2 2)}
k k
v L L L L L L L H
mC E T a T xT E a xT a xT T
π= ψ η ψ+ ⎢⎡⎣ ′′− ⎤ ⎡⎦ ⎣⎥ ⎢ − ′′− ⎥⎤⎦− ′′ψ η ψ+ − (27)
wherea′′ =kEH2 EL, 1 [(1 2) 2 ]
k
L H L H H
Equations (26) and (27) are the finite time exergoeconomic performance characteristic of a steady flow endoreversible Diesel refrigeration cycle.
WhenCout1=Cp, Cout2=Cv andCin=Cv, equations (15) and (21) become:
( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 k
L L L
k k
L H L L L L L
T a T xT
a xT T E a xT T a T xT
ε= ′′′ −
⎡ ′′′ ⎤ ′′′
′′′ − ⎢⎣ − − ⎥⎦− + (28)
( ) ( ) ( ) ( ) ( )
{
1 1 2 1 1 1 ( 1 2 2)}
k k
v L L L L L L L H
mC E T a T xT E a xT a xT T
π= ψ η ψ+ ⎢⎡⎣ ′′′− ⎤ ⎡⎦ ⎣⎥ ⎢ − ′′′− ⎥⎤⎦− ′′′ψ η ψ+ − (29)
wherea′′′ =⎣⎡kEH1+EH2(1−EH1)⎤⎦ EL, ( )
( )1 ( )
1 1 1 1 (1 1)(1 2) 1 2 1 2
k
H L H H H H L H H H H H
a′′′ =⎣⎡ −E xT +E T ⎦⎤ − ⎡⎣ −E −E xT + E +E −E E T ⎤⎦.
Equations (28) and (29) are the finite time exergoeconomic performance characteristic of a steady flow endoreversible Dual refrigeration cycle.
When Cin=Cout1=Cout2→ ∞, equations (15) and (21) are the finite time exergoeconomic performance
characteristic of the endoreversible Carnot refrigeration cycle [31, 38].
Equations (15) and (21) are the major performance relations for the endoreversible refrigeration cycle coupled to two constant-temperature reservoirs. They determine the relations between the COP and the temperature ratio of the working fluid, between the profit rate and the temperature ratio of the working fluid, as well as between the profit rate and the COP. Finding the optimum f (f =U UL H) with the
constraint of UH1+UH2+UL=UH+UL=UT, one may obtain the optimal profit rate (πopt) and the optimal COP
for the fixed temperature ratio of the working fluid. The optimal COP is a monotonically increasing function of the temperature ratio of the working fluid, while there exists a maximum profit rate for an optimal temperature ratio of the working fluid. Maximizing πopt with respect to x by setting ∂πopt ∂ =x 0 in
Eq. (21) yields the maximum profit rate πmax and the optimal temperature ratio of the working fluid xopt.
Furthermore, substituting xopt into equation (15) after optimizing UL UH yields εm, which is the finite-time
thermodynamic exergoeconomic bound.
The idea mentioned above may be applied to various endoreversible cycles, including Brayton cycle by setting Cin=Cout=CPor Otto cycle by setting Cin=Cout=Cv. For the endoreversible Brayton or Otto
refrigeration cycle, when UH=UL=UT 2, the profit rate approaches its optimum value for a given COP.
The relation between the optimal profit rate and COP is:
(
1)
( ) ( )1 1 2 1 2 2
1
1
1 {exp[ (2 )] 1} {exp[ (2 )] 1}
1
opt mC TL TH UT mC UT mC
π ε ψ η ψ ψ η ψ
ε − − ⎡ ⎤ ⎡ ⎤ = ⎣ + − ⎦ +⎢ + − + ⎥ − + ⎣ ⎦ (30)
Maximizing πopt with respect to ε by setting ∂πopt ∂ =ε 0 in Eq. (25) directly yields the maximum profit
rate and the corresponding optimal COP εm, that is, the finite-time thermodynamic exergoeconomic
bound:
( ) ( )
{
0.5}
{
( )( ) 0.5 ( )}
max 1 1 2 1 2 2 1 1 2 1 2 2 / 1 2 2
{exp[ (2 )] 1} {exp[ (2 )] 1}
H L H L H
T T
mC T T T T T
U mC U mC
π = ⎣⎡ ψ η ψ+ ψ η ψ+ ⎦⎤ − ⎡⎣ ψ η ψ ψ η ψ+ + ⎤⎦ −ψ η ψ+
− +
(31)
( ) ( )
{
}
{
0.5}
11 1 2 1 2 2 1
m TH TL
ε = ψ η ψ+ ⎡⎣ψ η ψ+ ⎤⎦ − − (32)
The finite-time thermodynamic exergoeconomic bound (εm) is different from the classical reversible
bound and the finite-time thermodynamic bound at the maximum cooling load output. It is dependent on
H
T , TL , T0 and ψ ψ2 1.
Note that for the process to be potential profitable, the following relationship must exist: 0<ψ ψ2 1<1,
because one unit of work input must give rise to at least one unit of exergy output. As the price of exergy output becomes very large compared with the price of the work input, i.e. ψ ψ2 1→0, equation (21)
1A
π ψ= (33)
That is the profit rate maximization approaches the exergy output maximization, where A is the rate of
exergy output of the universal endoreversible refrigeration cycle.
On the other hand, as the price of exergy output approaches the price of the work input, i.e. ψ ψ2 1→1,
equation (21) becomes
1 0T
π= −ψ σ (34)
where σ is the rate of entropy production of the universal endoreversible refrigeration cycle. That is the profit rate maximization approaches the entropy production rate minimization, in other word, the minimum waste of exergy. Equation (34) indicates that the refrigerator is not profitable regardless of the COP is at which the refrigerator is operating. Only the refrigerator is operating reversibly (ε ε= C) will the
revenue equal the cost, and then the maximum profit rate will equal zero. The corresponding rate of entropy production is also zero.
5. Numerical examples
To illustrate the preceding analysis, numerical examples are provided. In the calculations, it is set that
0 298.15
T = K, TH=T0, .
1.1165 /
m= kg s, EH=EL=0.9, cv=0.7166kJ/(kg K⋅ ), cp=1.0032kJ/(kg K⋅ ), and τ=TH TL=1.4. A
dimensionless profit rate is defined as Π =π (T E CL L Vψ2).
Figures 2-6 show the effects of the price ratio on the dimensionless profit rate versus temperature ratio of the working fluid and the COP versus temperature ratio of the working fluid for Otto, Diesel, Atkinson, Dual and Brayton refrigeration cycles. Figure 7 shows the effects of the price ratio on the dimensionless profit rate versus the COP for five cycles.
From the Figures 2-5, one can see that the COP decreases monotonically when x increases for any one
of the five cycles, while the profit rate versus x is parabolic-like one. When ψ ψ1/ 2=1.0, the profit rate
maximization approaches zero, this means that the refrigerator is not profitable in any case. From Figure 7, one can see that , when ψ ψ1/ 2=1.0, the profit rate approaches to zero as the COP increases; when
1 2 1
ψ ψ > , the curves of the dimensionless profit rate versus the COP are parabolic-like ones. The COP at the maximum profit rate is the finite-time exergoeconomic performance bound. Therefore, from the above analysis, one can find that the effect of the price ratio ψ ψ1 2 on the finite-time exergoeconomic
performance bound is larger: when ψ ψ1/ 2=1.0, the profit rate approaches to zero as the COP increases;
when ψ ψ1 21, the finite-time exergoeconomic performance bound of the endoreversible refrigerator
approaches to the finite-time thermodynamic performance bound. Therefore, the finite-time exergoeconomic performance bound (επ) lies between the finite-time thermodynamic performance bound and the reversible performance bound. επ is related to the latter two through the price ratio, and the associated COP bounds are the upper and lower limits of επ.
Figure 3. Dimensionless profit rate and the COP characteristic for Atkinson cycle
Figure 4. Dimensionless profit rate and the COP characteristic for Diesel cycle
Figure 5. Dimensionless profit rate and the COP characteristic for Dual cycle
Figure 7. Dimensionless profit rate versus the COP characteristic for five cycles
6. Conclusion
Economics plays a major role in the thermal power and cryogenics industry. This paper combines finite time thermodynamics with exergoeconomics to form a new analysis of universal endoreversible refrigeration cycle model. One seeks the economic optimization objective function instead of pure thermodynamic parameters by viewing the refrigerator as a production process. It is shown that the economic and thermodynamic optimization converged in the limits ψ ψ1 2→0 and ψ ψ1 2→1. Analysis and
optimization of the model are carried out in order to investigate the effect of cycle process on the performance of the cycles using numerical examples. The results obtained herein include the performance characteristics of endoreversible Carnot, Diesel, Otto, Atkinson, Dual and Brayton refrigeration cycles.
Acknowledgments
This paper is supported by The National Natural Science Foundation of P. R. China (Project No. 10905093), the Program for New Century Excellent Talents in University of P. R. China (Project No 20041006) and The Foundation for the Author of National Excellent Doctoral Dissertation of P. R. China (Project No. 200136).
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Lingen Chen received all his degrees (BS, 1983; MS, 1986, PhD, 1998) in power engineering and engineering thermophysics from the Naval University of Engineering, P R China. His work covers a diversity of topics in engineering thermodynamics, constructal theory, turbomachinery, reliability engineering, and technology support for propulsion plants. He has been the Director of the Department of Nuclear Energy Science and Engineering, the Superintendent of the Postgraduate School, and the President of the College of Naval Architecture and Power. Now, he is the President of the College of Power Engineering, Naval University of Engineering, P R China. Professor Chen is the author or co-author of over 1220 peer-refereed articles (over 560 in English journals) and nine books (two in English).
E-mail address: lgchenna@yahoo.com; lingenchen@hotmail.com, Fax: 0086-27-83638709 Tel: 0086-27-83615046
Xuxian Kan received all his degrees (BS, 2005; PhD, 2010) in power engineering and engineering thermophysics from the Naval University of Engineering, P R China. His work covers topics in finite time thermodynamics and thermoacoustic engines. He has published 10 research papers in the international journals.
Fengrui Sun received his BS Degrees in 1958 in Power Engineering from the Harbing University of Technology, P R China. His work covers a diversity of topics in engineering thermodynamics, constructal theory, reliability engineering, and marine nuclear reactor engineering. He is a Professor in the Department of Power Engineering, Naval University of Engineering, P R China. Professor Sun is the author or co-author of over 750 peer-refereed papers (over 340 in English) and two books (one in English).
